From: cardano
Subject: Long runs of winning and losing
Date: 4/9/97

In another thread (on stripped deck poker) Robert Copps poses the
question of the likelihood of long runs of winning, specifically the
chances of winning 29, 30, 31, or 32 times in a row.  Here are some
useful probabilities on this sort of thing:


In order to calculate the likelihood of exceptional runs of good or
bad luck in gambling it is necessary to use the Theory of Runs, usually
only covered in texts of advanced probability theory.  The equations
have no analytic solution and the results need to be calculated
numerically.  I wrote a program in Mathcad 4.0 to solve these problems.
I need only plug in the values of p, r, and N in order to get P,
where p = probability of an individual event
      r = number of consecutive occurrences of the event, or run length
      N = number of trials
      P = probability of run of length r or longer occurring

Assumptions:  events are independent of one another and probability
of individual events remains constant at p, and p is known.  In fact
in poker the events may not be completely independent and p is not
known, but must be estimated from past results.    

Some sample results:

Assume that a winning poker player wins 80% of his or her sessions.

Number of Sessions, N    500     750     1000
     r = 29              13.8    20.3    26.3
     r = 30              11.2    16.6    21.6
     r = 31               9.0    13.5    17.7
     r = 32               7.3    10.9    14.4

Numbers in table refer to percentage chance of run of length r
or longer.

Now for the dark side (consecutive losses):   :-(                         

Number of Sessions, N    500     750     1000     1500
     r = 4               47.3    61.8    72.3     85.5
     r = 5               11.9    17.4    22.5     31.9

Food for thought.  Of course, a small change in the winning rate
will have a large effect on the results.

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                  * Brent Fredrickson *
..After investigation, believe that which you have
yourself tested and found reasonable...    Kalama Sutra
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